Theorem riotasv2s 38937

Description: The value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5342) in the form of a substitution instance. Special case of riota2f 7330. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypothesis

Ref	Expression
riotasv2s.2	⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))

Assertion

Ref	Expression
riotasv2s	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝑥,𝐸,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasv2s

Step	Hyp	Ref	Expression
1		3simpc 1150	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → (𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)))
2		simp1 1136	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐴 ∈ 𝑉)
3		riotasv2s.2	. . . . . 6 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
4		nfra1 3253	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
5		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
6	4, 5	nfriota 7318	. . . . . 6 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
7	3, 6	nfcxfr 2889	. . . . 5 ⊢ Ⅎ𝑦𝐷
8	7	nfel1 2908	. . . 4 ⊢ Ⅎ𝑦 𝐷 ∈ 𝐴
9		nfv 1914	. . . . 5 ⊢ Ⅎ𝑦 𝐸 ∈ 𝐵
10		nfsbc1v 3762	. . . . 5 ⊢ Ⅎ𝑦[𝐸 / 𝑦]𝜑
11	9, 10	nfan 1899	. . . 4 ⊢ Ⅎ𝑦(𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)
12	8, 11	nfan 1899	. . 3 ⊢ Ⅎ𝑦(𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑))
13		nfcsb1v 3875	. . . 4 ⊢ Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶
14	13	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦⦋𝐸 / 𝑦⦌𝐶)
15	10	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → Ⅎ𝑦[𝐸 / 𝑦]𝜑)
16	3	a1i 11	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
17		sbceq1a 3753	. . . 4 ⊢ (𝑦 = 𝐸 → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
18	17	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → (𝜑 ↔ [𝐸 / 𝑦]𝜑))
19		csbeq1a 3865	. . . 4 ⊢ (𝑦 = 𝐸 → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
20	19	adantl 481	. . 3 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝑦 = 𝐸) → 𝐶 = ⦋𝐸 / 𝑦⦌𝐶)
21		simpl 482	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 ∈ 𝐴)
22		simprl 770	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐸 ∈ 𝐵)
23		simprr 772	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → [𝐸 / 𝑦]𝜑)
24	12, 14, 15, 16, 18, 20, 21, 22, 23	riotasv2d 38936	. 2 ⊢ (((𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) ∧ 𝐴 ∈ 𝑉) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)
25	1, 2, 24	syl2anc 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝐴 ∧ (𝐸 ∈ 𝐵 ∧ [𝐸 / 𝑦]𝜑)) → 𝐷 = ⦋𝐸 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  [wsbc 3742  ⦋csb 3851  ℩crio 7305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-riotaBAD 38932

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-riota 7306  df-undef 8206

This theorem is referenced by: (None)